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A bordism approach to string topology. (English) Zbl 1086.55004

Using classical intersection theory of chains in a closed manifold, Chas and Sullivan have defined new operations on the free loop space homology \(H_*(LM)\) of \(M\). These operations are called loop product, loop bracket, string bracket, etc... A purely homotopical presentation has been given by R. L. Cohen and J. D. S. Jones using a ring spectrum structure on a Thom spectrum of a virtual bundle over \(LM\), with, as a corollary, an isomorphism between the loop algebra \(H_*(LM)\) and the Hochschild cohomology of the cochain algebra on \(M\), see [Math. Ann. 324, 773–798 (2002; Zbl 1025.55005)].
The author gives a new approach to string topology: He uses the geometric version of singular homology theory introduced by M. Jakob [Manuscr. Math. 96, No. 1, 67–80 (1998; Zbl 0897.55004)]. The main operations of string topology are then redefined and studied in that context. In particular, he extends the Frobenius structure described by Cohen and Godin on string homology to a homological action of the space of Sullivan chord diagrams on the free loop space.

MSC:

55P35 Loop spaces
55N35 Other homology theories in algebraic topology
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